Machine learning has significantly enhanced the abilities of robots, enabling them to perform a wide range of tasks in human environments and adapt to our uncertain real world. Recent works in various machine learning domains have highlighted the importance of accounting for fairness to ensure that these algorithms do not reproduce human biases and consequently lead to discriminatory outcomes. With robot learning systems increasingly performing more and more tasks in our everyday lives, it is crucial to understand the influence of such biases to prevent unintended behavior toward certain groups of people. In this work, we present the first survey on fairness in robot learning from an interdisciplinary perspective spanning technical, ethical, and legal challenges. We propose a taxonomy for sources of bias and the resulting types of discrimination due to them. Using examples from different robot learning domains, we examine scenarios of unfair outcomes and strategies to mitigate them. We present early advances in the field by covering different fairness definitions, ethical and legal considerations, and methods for fair robot learning. With this work, we aim to pave the road for groundbreaking developments in fair robot learning.